gh-40850: fixing a bug in perfect_power

Fix #40846

### 📝 Checklist

- [x] The title is concise and informative.
- [x] The description explains in detail what this PR is about.
- [x] I have linked a relevant issue or discussion.
- [x] I have created tests covering the changes.

URL: #40850
Reported by: Frédéric Chapoton
Reviewer(s): Martin Rubey, Sahil Jain, Vincent Macri

Author: Release Manager

build/pkgs/configure/checksums.ini

@@ -1,3 +1,3 @@
 tarball=configure-VERSION.tar.gz
-sha1=dacd801ebf4c551ecfc93ae1263f4c0acc79154d
-sha256=3d0d90cd76c6a4b8773366e44b459244d6ccef764f69d33a3591e09d7014fc26
+sha1=768af12e19ba6d0b1f7baf5de900c98dfedd8622
+sha256=1b6e5d83d56a78754699f492c669ffae696349b4ad58e8f3225202dd0dadfb40

build/pkgs/configure/package-version.txt

@@ -1 +1 @@
-468cdc7e77405ca20d6adad73010322b84e33718
+07b01bf0103549b7b03c6b356e0aa797b82d5e37

src/sage/rings/integer.pyx

@@ -4963,6 +4963,12 @@ cdef class Integer(sage.structure.element.EuclideanDomainElement):
             (2, 2)
             sage: 256.perfect_power()
             (2, 8)
+
+        Test small powers of even numbers that are not a power of 2
+        (see :issue:`40846`)::
+
+            sage: (26**2).perfect_power()
+            (26, 2)
         """
         cdef long n
         # Fast PARI-free path
@@ -4974,7 +4980,7 @@ cdef class Integer(sage.structure.element.EuclideanDomainElement):
                 if not (n & 1):
                     if mpz_popcount(self.value) == 1:
                         return smallInteger(2), smallInteger(mpz_sizeinbase(self.value, 2) - 1)
-                if n < 1000:
+                elif n < 1000:
                     if _small_primes_table[n >> 1]:
                         return self, one